Skim and discard foam from surface of water. Partially cover, reduce heat to low, and simmer 30 to 35 minutes or until vegetables are tender and chicken is no longer pink. While chicken simmers, finely chop tarragon and parsley leaves.

Discard chicken skin. Divide chicken and vegetables among serving bowls; discard stems. Ladle 1 cup cooking liquid into each bowl, then season with pinch salt and pepper. Garnish with tarragon and parsley. Strain remaining liquid through sieve; refrigerate for another use.